# Find Pairs of Numbers That Satisfy an Equation With Two Unknowns and Enumerate Possibilities of Combinations of Two Variables

For equations involving two unknowns, there may only be one solution, depending on how much information is in the question. If one unknown is given to you, then there is only one option for the other unknown.

Sometimes there are many pairs of numbers that can solve one problem. A question might ask you to find the total number of different solutions.

Example 1 => If b – a = 15 and b represents the number 23, what does a represent?
This question now only has a as an unknown.
23 – a = 15.
We can also say that 23 – 15 = a
So a represents the number 8. a = 8

Example 2 => Two consecutive numbers multiply to give 56. What are these two numbers?
There are two unknowns but we know their product and that they are consecutive.
The factors of 56 are =>1 x 56, 2 x 28, 4 x 14, 7 x 8
7 and 8 are consecutive, so these are the unknown values.

Example 3 => Arnie is older than Mabel by seven years. If the total of their two ages is 21, how old are Arnie and Mabel?
Whatever Mabel’s age is, Arnie is 7 years older.
21 – 7 = 14. 14 divided by 2 is 7.
Answer => Mabel is 7 years old, and Arnie is 7 + 7 = 14 years old.

Example 4 => Janine goes to a cafe and orders 3 coffees and 2 teas for herself and her friends. Janine pays £12.40. How much could a coffee and a tea cost?
One example could be that a coffee costs £2.50 and a tea costs £2.45
3 x £2.50 = £7.50, 2 x £2.45 = £4.90
£4.90 + £7.50 = £12.40
Another example could be that a coffee costs £3.00 and a tea costs £1.70.
3 x £3.00 = £9.00, 2 x £1.70 = £3.40
£9.00 + £3.40 = £12.40
There are many other solutions!

Example 5 => f + g = 9
f and g are both positive whole numbers.How many solutions are there for this equation with two unknowns?
f = 1 and g = 8, f = 8 and g = 1, f = 2 and g = 7, f = 7 and g = 2, f = 3 and g = 6, f = 6 and g = 3, f = 4 and g = 5, f = 5 and g = 4
There are eight solutions. Zero is not a positive whole number.

Now try a few practice questions!

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